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Central limit theorem for bifurcating markov chains under L2-ergodic conditions

Part of: Limit theorems Parametric inference Nonparametric inference Markov processes

Published online by Cambridge University Press:  15 June 2022

S. Valère Bitseki Penda Open the ORCID record for S. Valère Bitseki Penda [Opens in a new window]  and
Jean-François Delmas
S. Valère Bitseki Penda*
Affiliation:
Université Bourgogne Franche-Comté, IMB, CNRS-UMR 5584
Jean-François Delmas*
Affiliation:
CERMICS, Ecole des Ponts
*
*Postal address: Université Bourgogne Franche-Comté, IMB, CNRS-UMR 5584, 9 avenue Alain Savary, 21078 Dijon Cedex, France. Email address: simeon-valere.bitseki-penda@u-bourgogne.fr
**Postal address: CERMICS, Ecole des Ponts, 6 et 8, avenue Blaise Pascal Cité Descartes, Champs-sur-Marne 77455, Marne-la-Vallée Cedex 2, France. Email address: jean-francois.delmas@enpc.fr
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Abstract

Bifurcating Markov chains (BMCs) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for additive functionals of BMCs under $L^2$ -ergodic conditions with three different regimes. This completes the pointwise approach developed in a previous work. As an application, we study the elementary case of a symmetric bifurcating autoregressive process, which justifies the nontrivial hypothesis considered on the kernel transition of the BMCs. We illustrate in this example the phase transition observed in the fluctuations.

Keywords

Bifurcating Markov chainsbifurcating autoregressive processbinary treesfluctuations for tree-indexed Markov chainsdensity estimation

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 62G05: Estimation 62F12: Asymptotic properties of estimators
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 4 , December 2022 , pp. 999 - 1031
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

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Bitseki Penda, S. V. and Delmas, J.-F. (2021). Central limit theorem for bifurcating Markov chains under pointwise ergodic conditions. Preprint. Available at https://arxiv.org/abs/2012.04741v2.Google Scholar
Bitseki Penda, S. V. and Delmas, J.-F. (2022). Central limit theorem for bifurcating Markov chains under $L^{2}$ -ergodic conditions. Supplementary material. Available at https://doi.org/10.1017/apr.2022.3.Google Scholar
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